Embeddings between weighted Copson and Cesàro function spaces

Amiran Gogatishvili; Rza Mustafayev; Tuğçe Ünver

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 4, page 1105-1132
  • ISSN: 0011-4642

Abstract

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In this paper, characterizations of the embeddings between weighted Copson function spaces Cop p 1 , q 1 ( u 1 , v 1 ) and weighted Cesàro function spaces Ces p 2 , q 2 ( u 2 , v 2 ) are given. In particular, two-sided estimates of the optimal constant c in the inequality d ( ∫ 0 ∞ ∫ 0 t f ( τ ) p 2 v 2 ( τ ) d τ q 2 / p 2 u 2 ( t ) d t ) 1 / q 2 ≤ c ∫ 0 ∞ ∫ t ∞ f ( τ ) p 1 v 1 ( τ ) d τ q 1 / p 1 u 1 ( t ) d t 1 / q 1 , d where p 1 , p 2 , q 1 , q 2 ∈ ( 0 , ∞ ) , p 2 ≤ q 2 and u 1 , u 2 , v 1 , v 2 are weights on ( 0 , ∞ ) , are obtained. The most innovative part consists of the fact that possibly different parameters p 1 and p 2 and possibly different inner weights v 1 and v 2 are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.

How to cite

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Gogatishvili, Amiran, Mustafayev, Rza, and Ünver, Tuğçe. "Embeddings between weighted Copson and Cesàro function spaces." Czechoslovak Mathematical Journal 67.4 (2017): 1105-1132. <http://eudml.org/doc/294531>.

@article{Gogatishvili2017,
abstract = {In this paper, characterizations of the embeddings between weighted Copson function spaces $\{\rm Cop\}_\{p_1,q_1\}(u_1,v_1)$ and weighted Cesàro function spaces $\{\rm Ces\}_\{p_2,q_2\}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality \[ \begin\{aligned\}d \biggl ( \int \_0^\{\infty \} &\biggl ( \int \_0^t f(\tau )^\{p\_2\}v\_2(\tau ) \{\rm d\}\tau \biggr )^\{\{q\_2/p\_2\}\} u\_2(t) \{\rm d\} t\biggr )^\{\{1/q\_2\}\}\\ & \le c \biggl ( \int \_0^\{\infty \} \biggl ( \int \_t^\{\infty \} f(\tau )^\{p\_1\} v\_1(\tau ) \{\rm d\}\tau \biggr )^\{\{q\_1/p\_1\}\} u\_1(t) \{\rm d\} t\biggr )^\{\{1/q\_1\}\}, \end\{aligned\}d \] where $p_1,p_2,q_1,q_2 \in (0,\infty )$, $p_2 \le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.},
author = {Gogatishvili, Amiran, Mustafayev, Rza, Ünver, Tuğçe},
journal = {Czechoslovak Mathematical Journal},
keywords = {Cesàro and Copson function spaces; embedding; iterated Hardy inequalities},
language = {eng},
number = {4},
pages = {1105-1132},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Embeddings between weighted Copson and Cesàro function spaces},
url = {http://eudml.org/doc/294531},
volume = {67},
year = {2017},
}

TY - JOUR
AU - Gogatishvili, Amiran
AU - Mustafayev, Rza
AU - Ünver, Tuğçe
TI - Embeddings between weighted Copson and Cesàro function spaces
JO - Czechoslovak Mathematical Journal
PY - 2017
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 4
SP - 1105
EP - 1132
AB - In this paper, characterizations of the embeddings between weighted Copson function spaces ${\rm Cop}_{p_1,q_1}(u_1,v_1)$ and weighted Cesàro function spaces ${\rm Ces}_{p_2,q_2}(u_2,v_2)$ are given. In particular, two-sided estimates of the optimal constant $c$ in the inequality \[ \begin{aligned}d \biggl ( \int _0^{\infty } &\biggl ( \int _0^t f(\tau )^{p_2}v_2(\tau ) {\rm d}\tau \biggr )^{{q_2/p_2}} u_2(t) {\rm d} t\biggr )^{{1/q_2}}\\ & \le c \biggl ( \int _0^{\infty } \biggl ( \int _t^{\infty } f(\tau )^{p_1} v_1(\tau ) {\rm d}\tau \biggr )^{{q_1/p_1}} u_1(t) {\rm d} t\biggr )^{{1/q_1}}, \end{aligned}d \] where $p_1,p_2,q_1,q_2 \in (0,\infty )$, $p_2 \le q_2$ and $u_1$, $u_2$, $v_1$, $v_2$ are weights on $(0,\infty )$, are obtained. The most innovative part consists of the fact that possibly different parameters $p_1$ and $p_2$ and possibly different inner weights $v_1$ and $v_2$ are allowed. The proof is based on the combination of duality techniques with estimates of optimal constants of the embeddings between weighted Cesàro and Copson spaces and weighted Lebesgue spaces, which reduce the problem to the solutions of iterated Hardy-type inequalities.
LA - eng
KW - Cesàro and Copson function spaces; embedding; iterated Hardy inequalities
UR - http://eudml.org/doc/294531
ER -

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